On the group of spheromorphisms of a homogeneous non-locally finite tree

We consider a tree $\mathbb{T}$ all whose vertices have countable valency. Its boundary is the Baire space $\mathbb{B}\simeq\mathbb{N}^\mathbb{N}$ and the set of irrational numbers $\mathbb{R}\setminus\mathbb{Q}$ is identified with $\mathbb{B}$ by continued fraction expansions. Removing $k$ edges from $\mathbb{T}$, we get a forest consisting of copies of $\mathbb{T}$. A spheromorphism (or hierarchomorphism) of $\mathbb{T}$ is an isomorphism of two such subforests regarded as a transformation of $\mathbb{T}$ or $\mathbb{B}$. We denote the group of all spheromorphisms by $\operatorname{Hier}(\mathbb{T})$. We show that the correspondence $\mathbb{R}\setminus \mathbb{Q}\simeq \mathbb{B}$ sends the Thompson group realized by piecewise $\mathrm{PSL}_2(\mathbb{Z})$-transformations to a subgroup of $\operatorname{Hier}(\mathbb{T})$. We construct some unitary representations of $\operatorname{Hier}(\mathbb{T})$, show that the group $\operatorname{Aut}(\mathbb{T})$ of automorphisms is spherical in $\operatorname{Hier}(\mathbb{T})$ and describe the train (enveloping category) of $\operatorname{Hier}(\mathbb{T})$.

Thompson group, continued fraction, Baire space, representation of categories, Bruhat–Tits tree